The reflexive Banach space fixed point property conjecture

Let $X$ be a Banach space. It has the fixed point property for nonexpansive mappings when every nonempty closed, convex and bounded subset $C\subset X$ and every map $T\colon C\to C$ satisfying

$$\|T(x)-T(y)\|\leq\|x-y\|$$

for all $x,y\in C$ has a fixed point in $C$.

Reflexive-space fixed point conjecture. Every reflexive Banach space has the fixed point property for nonexpansive maps. Equivalently, if $X$ is reflexive and $C\subset X$ is closed, bounded and convex, then every nonexpansive self-map $T\colon C\to C$ has a fixed point.

Reflexivity guarantees weak compactness of bounded subsets, but known fixed point theorems generally require additional hypotheses such as uniform convexity or normal structure. Nonreflexive examples exist both with and without the property, while the existence of a reflexive Banach space failing the fixed point property remains open.